The Hidden Worlds of Quantum Mechanics

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The Hidden Worlds of Quantum Mechanics

• Craig Callender
• Philosophy, UCSD
• ccallender_at_ucsd.edu

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Double Slit Experiment
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(No Transcript)
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What you would expect is
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But what actually happens is

• 1961, Jönsson, Zeitschrift für Physik 161 454
• 1974, P. Merli, G. Missiroli and G. Pozzi in
  Bologna in 1974
• Hitachi (A Tonomura et al). 1989 Demonstration of
  single-electron buildup of an interference
  pattern Am. J. Phys. 57

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The Double Slit Experiment
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Actual images
Bologna 1974
Hitachi 1989
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(No Transcript)
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Structure of Physical Theories

• Stuff Newtonian corpuscles
• State (xi, pi)
• Dynamical law Hamiltons equations

• Stuff ?
• State wavefunction or vector ?gt
• Dynamical law Schrodingers equation

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Wave function (quantum state)
?(x)
x
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?2(x) gives the probability of finding the
particle at position x
Most probable location of particle
?2(x)
x
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where the observable is assumed to have a
discrete spectrum of eigenvalues, u are the
(normalized) eigenfunctions, and the coefficient
cn of the nth term gives the probability of the
nth eigenvalue via cn2.
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Schrodinger evolution
Schrodinger evolution
?i(x)
?f(x)

• Deterministic
• Unitary
• Linear

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Three Ingredients for Trouble

• Linear dynamical evolution
• Eigenstate-eigenvalue rule
• Determinate outcomes

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Linearity

• If the evolution takes Agt ?Bgt..
• And takes Cgt ? Dgt
• Then it takes the state
• Agt Cgt ? Bgt Dgt

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Eigenstate-eigenvalue link

• A system in the quantum state ?gt has the value a
  for the observable  if and only if ?gt assigns
  the probability 1 to a and the probability 0 to
  all other possible values of Â.
• Â ?gt a?gt

?(x)
x
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Measurement

• 1 readygtM?gtS ? upgtM?gtS
• 2 readygtM?gtS ? downgtM?gtS
• readygtM (a1?gtSa2?gtS)
• a1readygtM?gtSa2readygtM?gtS
• ? (a1upgtM?gtS a2downgtM?gtS)

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Schrödingers cat

• 1 cat readygtreadygtM?gtS ? cat deadgtupgtM?gtS
  
• 2 cat readygtreadygtM?gtS ? cat
  alivegtdowngtM?gtS
• cat readygtreadygtM (a1?gtS a2?gtS)
• a1cat readygtreadygtM?gtS a2cat
  readygtreadygtM?gtS
• ? (a1deadgtupgtM?gtS a2alivegtdowngtM?gtS

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Measurement Problem

• The quantum state is representationally complete,
  i.e., the eigenstate-eigenvalue link holds
• The quantum state always evolves according to a
  linear laws of evolution, e.g., Schrodinger
  equation.
• Measurements yield definite values
• Contradiction!

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Schrodinger evolution
?f(x)
?i(x)
Final state is a probability distribution but in
the real world something actually happens!
non-Schrodinger evolution miracle collapse
?i(x)
?f(x)
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MP is Here to Stay

• Quantum field theory employs superpositions among
  distinct macroscopic states
• Theories on the horizon do too, e.g., superstring
  theory, loop theory, etc.
• So MP has to be solved

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The Standard Solution

• Copenhagen (Bohr, Heisenberg, Dirac, von Neumann,
  )
• Two types of evolution
• Unmeasured evolution
• Measurement evolution
• Quantum philosophy of realism about
  macroscopic entities but anti-realism about
  microscopic ones

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Unique?

• Rosenfeld quantum theory eminently possess this
  character of uniqueness every feature of it has
  been forced upon us as the only way to avoid the
  ambiguities which would essentially affect any
  attempt at an analysis in classical terms of
  typical quantum phenomena

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Criticism

• It would seem that the theory is exclusively
  concerned about "results of measurement", and has
  nothing to say about anything else. What exactly
  qualifies some physical systems to play the role
  of "measurer"? Was the wavefunction of the world
  waiting to jump for thousands of millions of
  years until a single-celled living creature
  appeared? Or did it have to wait a little longer,
  for some better qualified system ... with a
  Ph.D.? If the theory is to apply to anything but
  highly idealized laboratory operations, are we
  not obliged to admit that more or less
  "measurement-like" processes are going on more or
  less all the time, more or less everywhere.

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Solution Deny one of the premises

1. The quantum state is representationally complete,
   i.e., the eigenstate-eigenvalue link holds
2. The quantum state always evolves according to a
   linear laws of evolution, e.g., Schrodinger
   equation.
3. Measurements yield definite values

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Deny 1

• Bohm-like hidden variable theories
• De Broglie 1927
• Bohm 1952
• Bohm and Vigier
• Nelsons stochastic mechanics
• Bell 1987
• Goldstein, Durr and Zanghi 1991

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Deny 2

• Physically-specifiable Collapse theories
• Pearle 1989
• Pearle and Squires 1994
• Pearle 1996
• Ghiradhi, Rimini and Weber 1986
• Bell 1987
• Penrose

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Deny 3

• Many-world type theories
• Everett 1957
• deWitt
• Many minds
• Barbour
• Rovellis relational qm

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Bohmian Mechanics

• De Broglie 1927 David Bohm 1952
• The de Broglie-Bohm idea seems so natural and
  simple, to resolve the wave-particle dilemma in
  such a clear and ordinary way, that it is a great
  mystery that it was so generally ignored. Bell,
  1987.

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Bohmian Mechanics

• Basic idea Suppose that there are some particles
  and that their velocities are determined by ?
  In other words, ? is not the whole story there
  are also particles.

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Bohmian mechanics
Schrodinger equation
Velocity equation
at time t0
If
at time t
then
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GRW
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• The world is described by two equations, the
  Schrodinger equation and the velocity equation.
  The latter is arguably the simplest first order
  equation for the positions of particles
  compatible with the Galilean and time reversal
  invariance of the former.
• Given any probability distribution for the
  initial configuration, Bohmian mechanics defines
  a probability distribution for the full
  trajectory. Notice that the velocity equation is
  simply v J/p, where J is the quantum probability
  current and p is the quantum probability density.
  It follows from the quantum continuity equation
  that if the distribution of the configuration Q
  is given by psi at some time (say the initial
  time) this will be true at all times.
• This deterministic theory of particles completely
  accounts for all the phenomena of nonrelativistic
  quantum mechanics, from interference effects to
  spectral lines. Thus Bohmian mechanics provides
  us with probabilities for complete
  configurational histories that are consistent
  with the quantum mechanical probabilities for
  configurations, including the positions of
  measuring devices.

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Many Bohmian Theories

• Deterministic alternatives to the velocity
  equation that are empirically adequate
• Indeterministic versions of velocity equation
• Spin as fundamental with position (Bohm, Schiller
  and Tiomno 1955, Dewdney 1992, Holland and Vigier
  1988, Bohm and Hiley 1993)
• Bell-Bub-Vink dynamics discrete, indeterministic
• Bub modal interpretation

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Ghirardi, Rimini Weber 1986

• Basic Idea The Schrodinger equation is not quite
  right. The wavefunction ?(r1,r2, ..., t) usually
  evolves according to the Schrodinger equation,
  but every now and then, at random, ? is
  multiplied (hit) by a gaussian function (and
  then normalized). How often the state is likely
  to be hit by a gaussian is proportional to how
  many particles there are in the system.
• The effect of this multiplication is to collapse
  the state to a more localized one. Thus, systems
  with large N turn out to be overwhelmingly likely
  to collapse, and systems with small N turn out to
  be unlikely to collapse.

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GRW
?(x)
x
Cat Alive
Cat Dead
1/?2(?C(r1,r2, ...rN...rM, t)??S(r1))
(?C(r1,r2, ...rN...rM, t)?? S(r1))
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Then the Gaussian hits
?(x)
j(x) K exp (-x - ri2/2a2)
? j(x, ri)?(...,t)/ Ri(x).
x
Cat Alive
a10-5cm, jump time T1016s For N 1023,
collapse happens around 10-7s, compared to
observation time 10-2s
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Many GRWs

• Ghirardi emphasizes the importance of specifying
  what he calls the physical reality of what
  exists out there.''
• Mass density interpretation for the simple GRW
  theory described here can be identified with the
  mass weighted sum, over all particles, of the
  one-particle densities arising from integrating
  over the coordinates of all but one of the
  particles.
• Hit interpretation Bell p 205, that the
  space-time points (x,t) at which the hits are
  centered (which are determined by the wave
  function trajectory) should themselves serve as
  the local beables of the theory. These are the
  mathematical counterparts in the theory to real
  events at definite places and times in the real
  world (as distinct from the many purely
  mathematical constructions that occur in the
  working out of physical theories, as distinct
  from things which may be real but not localized,
  and as distinct from the observables' of other
  formulations of quantum mechanics, for which we
  have no use here.) A piece of matter then is a
  galaxy of such events.'
• SL v CSL

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Bohm GRW Everett
Deterministic? Yes No Yes
Time reversal invariant? Yes No Yes
Monistic? No Yes Yes
Particles? Yes No No
Is spin real? No Yes Yes
Preferred foliation? Yes Yes Maybe not
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• There is something wrong with all of these
  post-Copenhagen interpretationsthey dont offer
  new predictions
• But
• Why should when the theory is developed be
  important?
• De Broglie 1927
• GRW will differ from Copenhagen Bohm would if
  Copenhagen were clear plus, one never knows

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Underdetermination of Theory by Evidence
THEORY1
THEORY2
Observable evidence
THEORY3
THEORY4
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• Duhem
• Shall we ever dare to assert that no other
  hypothesis is imaginable? Light may be a swarm
  of projectiles, or it may be a vibratory motion
  whose waves are propagated in a medium is it
  forbidden to be anything else at all?
• (1914)

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Experimenta Crucis?

• Bohm v Copenhagen
• Times of arrival, etc.
• Conroversial
• Bohm v Everett
• in principle underdetermination
• Laudan and Leplin 1991
• GRW v Bohm (or Copenhagen)
• Localizations ? greater KE ? heating
• Resistance of a superconductor different (Gallis
  Flemming Rae Rimini)
• One mole of H one atom excited/sec
• hope of reaching a crucial testextremely dim

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Theoria crucis?

• Solving the measurement problem seems to involve
  adding some new physics
• The new physics may or may not be experimentally
  detectable in the future
• But it might be crucial to new theories, e.g.,
  Bohmian quantum gravity, GRWs energy
  contribution, and so on.

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Philosophers are any real theories
under-determined?

• Our evidence is equally compatible with T (our
  best physical theory) and T (we and our
  apparently T-governed world is a computer
  simulation)
• But T is just skepticism, not a real physical
  theory

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Philosophers underdetermination too local to be
interesting?
The evidence equally supports T (newtonian
mechanics plus gravitational theory plus universe
is at rest in absolute space) and T (same, but
universe moving 5mph wrt absolute space)
5mph
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How should we react in QM case?
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Lesson of Quantum Mechanics Bad News

• GRW, Bohm, Everett, etc. show that there is
  underdetermination by genuine scientific
  theories, contrary to what some philosophers
  suggest.
• Furthermore, its hardly too local to care
  aboutit involves the central terms of our most
  fundamental theory
• Real life is stranger than fiction or philosophy

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Good news

• Further testing may narrow down the available
  possibilities
• Further theorizing may narrow down the
  possibilities
• The UT doesnt seem to be of the kind that would
  challenge realist interpretations of most
  science.
• First, one might appeal to non-observable facts,
  e.g., simplicity
• Second, the underdetermination is not entirely
  general
• Third, there are still levels that are not
  under-determined wrt the evidence, e.g., energy
  nuclear levels (Cordero 2002), scattering, etc.
• Nevertheless, were stuck with a bewildering
  amount of underdetermination, like it or notso
  its best to learn to like it.